In ∆ le ABC, cot(A/2)+cot(B/2)+cot(C/2)cot A+cot B+cot C=

∑cot(A/2)=∑ss-as-bs-c=∑ss-a∆=s2∆. ∑cot A=∑cos Asin A=∑2bc cos A2bc sin A=∑b2+c2-a24 ∆=a2+b2+c24∆ G.E.=s2∆a2+b2+c2/4∆ =4s2a2+b2+c2=(2s)2a2+b2+c2=a+b+c2a2+b2+c2.

In ∆ le ABC, cot(A/2)+cot(B/2)+cot(C/2)cot A+cot B+cot C=

∑cot(A/2)=∑ss-as-bs-c=∑ss-a∆=s2∆. ∑cot A=∑cos Asin A=∑2bc cos A2bc sin A=∑b2+c2-a24 ∆=a2+b2+c24∆ G.E.=s2∆a2+b2+c2/4∆ =4s2a2+b2+c2=(2s)2a2+b2+c2=a+b+c2a2+b2+c2.

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In $∆ l e ABC, \frac{\cot (A / 2) + \cot (B / 2) + \cot (C / 2)}{\cot A + \cot B + \cot C} =$

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$∆$

$\frac{a^{2} + b^{2} + c^{2}}{{(a + b + c)}^{2}}$

$\frac{{(a + b + c)}^{2}}{a^{2} + b^{2} + c^{2}}$

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Detailed Solution

$\sum \cot (A / 2) = \sum \sqrt{[\frac{s (s - a)}{(s - b) (s - c)}]} = \sum \frac{s (s - a)}{∆} = \frac{s^{2}}{∆} . \sum \cot A = \sum \frac{\cos A}{\sin A} = \sum \frac{2 bc \cos A}{2 bc \sin A} = \sum \frac{b^{2} + c^{2} - a^{2}}{4_{} ∆} = \frac{a^{2} + b^{2} + c^{2}}{4 ∆} G . E . = \frac{\frac{s^{2}}{∆}}{(a^{2} + b^{2} + c^{2}) / 4 ∆_{}} = \frac{4 s^{2}}{a^{2} + b^{2} + c^{2}} = \frac{{(2 s)}^{2}}{a^{2} + b^{2} + c^{2}} = \frac{{(a + b + c)}^{2}}{a^{2} + b^{2} + c^{2}} .$